Charles Pugh

\begin{conjecture} Let $f\in Diff^{r}(M)$ and $p\in\omega_{f} $. Then for any neighborhood $V_{f}\subset Diff^{r}(M) $ there is $g\in V_{f}$ such that $p$ is periodic point of $g$ \end{conjecture} There is an analogous conjecture for flows ( $C^{r}$ vector fields . In the case of diffeos this was proved by Charles Pugh for $r = 1$. In the case of Flows this has been solved by Sushei Hayahshy for $r = 1$ . But in the two cases the problem is wide open for $r > 1$